When children start to leam to read English, they benefit from leaming grapheme-phoneme correspondences. As they become more skilled, they use larger graphophonic units and morphemes in word recognition and spelling. We hypothesized that these 2 types of units in decoding make independent contributions to children's reading comprehension and fluency and that the use of morphological units is the stronger predictor of both measures. In a longitudinal study with a large sample in the United Kingdom, we tested through multiple regressions the contributions that these different units make to the prediction of reading competence (reading comprehension and fluency). The predictors were measured when the children were aged 8-9 years. Reading comprehension and rate were measured concurrentiy, and reading list fluency was measured at 12 and 13 years. After controlling for age and verbal IQ, the children's use of larger graphophonic units and their use of morphemes in reading and spelling made independent contributions to predicting their reading comprehension and reading fluency. The use of morphemes was the stronger predictor in all analyses. Thus, teaching that promotes the development of these different ways of reading and spelling words should be included in policy and practice.The aim of the study reported here is to analyze the effect of different decoding processes on reading competence. We distinguish two types of units that are used by children in decoding: graphophonic units, which carry no meaning, and morphological units, which are units of meaning. We suggest that this analysis of decoding processes can contdbute new insights to the understanding of reading competence. We consider reading competence as assessed by measures of both comprehension and fluency, in view of the fact that fiuency is considered a significant factor connecting

There is clear justification for making a distinction between mathematical reasoning and arithmetic skills. The implication is that schools must plan explicitly to improve mathematical reasoning as well as arithmetic skills.

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The basis of this intervention study is a distinction between numerical calculus and relational calculus. The former refers to numerical calculations and the latter to the analysis of the quantitative relations in mathematical problems. The inverse relation between addition and subtraction is relevant to both kinds of calculus, but so far research on improving children's understanding and use of the principle of inversion through interventions has only been applied to the solving of a+b−b=? sums. The main aim of the intervention described in this article was to study the effects of teaching children about the explicit use of inversion as part of the relational calculus needed to solve inverse addition and subtraction problems using a calculator. The study showed that children taught about relational calculus differed significantly from those who were taught numerical procedures, and also that effects of the intervention were stronger when children were taught about relational calculus with mixtures of indirect and direct word problems than when these two types of problem were given to them in separate blocks.Keywords Inverse relation between addition and subtraction . Relational calculus . Numerical calculus . Teaching the inverse relation Our aim in this study is to evaluate two different approaches to teaching children about the inverse relation between addition and subtraction in the context of story problems. The teaching of arithmetic must consider two different questions about the types of cognitive demand that children face in solving problems. One is how they can efficiently and accurately carry out arithmetical calculations. The second is how they know the right arithmetical operation to use in order to solve a particular problem.Vergnaud (1979) distinguishes these two abilities by referring to the first as "numerical calculus" and to the second as "relational calculus". By numerical calculus, Vergnaud means the operations on the numbers to find the sum, in an addition, or the rest, in a Educ Stud Math (2012) 79:371-388